Design: Blocked RCT, with 2 levels, and randomization done at level 1 (individual level).

Models: Constant treatment effects, fixed treatment effects, and random treatment effects.

Power Validation

Default parameters:

  • M = 3
  • J = 20
  • rho: \(\rho\) = 0.5
  • ATE ES = 0.125, 0.125, 0.125
  • R2.1: \(R^2_1\) = 0.1, 0.1, 0.1

Parameters by model type:

  • Omega: \(\omega_2 = 0\) for constant effects, \(\omega_2\) = for fixed and random
  • ICC: \(\text{ICC}_2 = 0\) for constant and fixed effects, \(\text{ICC}_2\) = 0.2, 0.2, 0.2 for random effects

Assumptions:

  • Two-level design: \(\text{ICC}_3 = 0\), \(\omega_3 = 0\), \(K = 1\)

Varying school size

\(\bar{n}\) = 100

\(\bar{n}\) = 75

\(\bar{n}\) = 50

Varying R2

\(R^2_1 =\) 0.6, 0.6, 0.6

\(R^2_2 =\) 0.6, 0.6, 0.6 (This should not affect power!)

\(R^2_1 =\) 0, 0, 0 \(R^2_2 =\) 0, 0, 0

Varying rho

\(\rho =\) 0.2

\(\rho =\) 0.8

Varying true positives

ATE (ES) = 0.125, 0, 0

Varying ICC

\(\text{ICC}_2\) = 0.8, 0.8, 0.8

MDES validation

## 
## 
## +------------+---------------+-------------+-------------+
## |    MTP     | Adjusted MDES | Indiv Power | Target MDES |
## +============+===============+=============+=============+
## |    rawp    |     0.126     |    0.555    |    0.125    |
## +------------+---------------+-------------+-------------+
## | Bonferroni |     0.122     |    0.36     |    0.125    |
## +------------+---------------+-------------+-------------+
## |     BH     |     0.126     |    0.488    |    0.125    |
## +------------+---------------+-------------+-------------+
## |    Holm    |     0.124     |    0.411    |    0.125    |
## +------------+---------------+-------------+-------------+
## 
## Table: blocked_i1_2c
## 
## 
## +------------+---------------+-------------+-------------+
## |    MTP     | Adjusted MDES | Indiv Power | Target MDES |
## +============+===============+=============+=============+
## |    rawp    |     0.127     |    0.559    |    0.125    |
## +------------+---------------+-------------+-------------+
## | Bonferroni |     0.123     |    0.364    |    0.125    |
## +------------+---------------+-------------+-------------+
## |     BH     |     0.125     |    0.487    |    0.125    |
## +------------+---------------+-------------+-------------+
## |    Holm    |     0.123     |    0.406    |    0.125    |
## +------------+---------------+-------------+-------------+
## 
## Table: blocked_i1_2f
## 
## 
## +------------+---------------+-------------+-------------+
## |    MTP     | Adjusted MDES | Indiv Power | Target MDES |
## +============+===============+=============+=============+
## |    rawp    |     0.124     |    0.458    |    0.125    |
## +------------+---------------+-------------+-------------+
## | Bonferroni |     0.118     |    0.234    |    0.125    |
## +------------+---------------+-------------+-------------+
## |     BH     |     0.126     |    0.382    |    0.125    |
## +------------+---------------+-------------+-------------+
## |    Holm    |     0.125     |    0.299    |    0.125    |
## +------------+---------------+-------------+-------------+
## 
## Table: blocked_i1_2r

Sample size validation

## 
## 
## +------+-------------+-------------+------------+--------------------+
## | MTP  | Sample type | Sample size | min2 power | Target sample size |
## +======+=============+=============+============+====================+
## |  BH  |      J      |     21      |   0.493    |         20         |
## +------+-------------+-------------+------------+--------------------+
## | Holm |      J      |     21      |   0.427    |         20         |
## +------+-------------+-------------+------------+--------------------+
## |  BH  |    nbar     |     51      |   0.488    |         50         |
## +------+-------------+-------------+------------+--------------------+
## | Holm |    nbar     |     51      |   0.429    |         50         |
## +------+-------------+-------------+------------+--------------------+
## 
## Table: blocked_i1_2c
## 
## 
## +------+-------------+-------------+------------+--------------------+
## | MTP  | Sample type | Sample size | min2 power | Target sample size |
## +======+=============+=============+============+====================+
## |  BH  |      J      |     20      |   0.477    |         20         |
## +------+-------------+-------------+------------+--------------------+
## | Holm |      J      |     20      |   0.408    |         20         |
## +------+-------------+-------------+------------+--------------------+
## |  BH  |    nbar     |     51      |   0.489    |         50         |
## +------+-------------+-------------+------------+--------------------+
## | Holm |    nbar     |     51      |   0.421    |         50         |
## +------+-------------+-------------+------------+--------------------+
## 
## Table: blocked_i1_2f
## 
## 
## +------+-------------+-------------+------------+--------------------+
## | MTP  | Sample type | Sample size | min2 power | Target sample size |
## +======+=============+=============+============+====================+
## |  BH  |      J      |     21      |   0.373    |         20         |
## +------+-------------+-------------+------------+--------------------+
## | Holm |      J      |     21      |   0.302    |         20         |
## +------+-------------+-------------+------------+--------------------+
## |  BH  |    nbar     |     52      |   0.375    |         50         |
## +------+-------------+-------------+------------+--------------------+
## | Holm |    nbar     |     50      |   0.295    |         50         |
## +------+-------------+-------------+------------+--------------------+
## 
## Table: blocked_i1_2r